Work of Darmon???
arXiv:0910.3875 Real multiplication and modular curves from arXiv Front: math.NT by Igor Nikolaev We construct an inverse of functor F, which maps isomorphism classes of elliptic curves with complex multiplication to the stable isomorphism classes of the so-called noncommutative tori with real multiplication. The construction allows to prove, that complex and real multiplication are mirror symmetric, i.e. F maps each imaginary quadratic field of discriminant -D to the real quadratic field of discriminant D.
arXiv:0912.4905 On a noncommutative reciprocity law from arXiv Front: math.NT by Igor Nikolaev We prove a reciprocity relation, which says that an L-function of the noncommutative torus with real multiplication coincides with the Hasse-Weil L-function of an elliptic curve with complex multiplication. Our proof is based on an explicit formula for the Teichmueller functor between elliptic curves and noncommutative tori. The result entails, that the Cuntz-Krieger algebras are isomorphic to elliptic curves over finite fields.
arXiv:1104.0609 On the rank conjecture from arXiv Front: math.NT by Igor Nikolaev The rank conjecture says that rank of the elliptic curve with complex multiplication is by one less the arithmetic complexity of the corresponding noncommutative torus with real multiplication. It is proved, that the conjecture is true for infinitely many pairwise non-isomorphic curves of rank zero.
nLab page on Real multiplication